<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">f</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">F</font> being  <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font><br/> for <font color="Olive">a</font>, <font color="Olive">x</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>   st <font color="Olive">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; <font color="Olive">F</font>,<font color="Olive">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Olive">f</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">x</font> holds <br/><font color="Olive">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Olive">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Olive">x</font>;<br/>
<a NAME="E1:2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:2_1"/>
<font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:2_1"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:2_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E4:2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">F</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:2_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:2_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span>
 <b>and </b><br/><a NAME="E7:2_1"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E8:2_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E9:2_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>hence </b><a NAME="E10:2_1"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E2:2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:2_2"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E4:2_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E5:2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">F</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:2_2"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E7:2_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:2_2"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc2.html#T36">MESFUNC2:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E13:2_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>
<a NAME="E14:2_2"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p0">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:2_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:2_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:2_2_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p0">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, , <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:2_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i><a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E16:2_2"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_2"><b>proof </b></a><div class="add">

<i><font color="Green">E50</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/>
<a NAME="E2:2_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E3:2_2_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:2_2_3"/>
 ex <font color="Olive">F1'</font> being  <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> st <br/>(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">F1'</font></span>)</span> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">F1'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_3_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">F1'</font> = <font color="Maroon">F1</font>;<br/>

<a NAME="E1:2_2_3_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:2_2_3_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:2_2_3_1_1"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:2_2_3_1_1"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E4:2_2_3_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E5:2_2_3_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:2_2_3_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E7:2_2_3_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:2_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_2_3_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1'</font></span>)</span> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_2_3"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:2_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:2_2_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E3:2_2_4"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_4"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E59</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E20:2_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<a NAME="E2:2_2_6"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E3:2_2_6"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:2_2_6"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E5:2_2_6"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:2_2_6"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E7:2_2_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:2_2_6"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="mesfunc4.html#T1" target="_self">Th1</a>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:2_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">F1</font>,<font color="Maroon">a1</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<i><font color="Green">E65</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> holds <br/><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<a NAME="E2:2_2_8_1_1_1"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:2_2_8_1_1_1"/><b>then </b>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_8_1_1_1"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:2_2_8_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> holds <br/><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<a NAME="E2:2_2_8_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:2_2_8_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:2_2_8_1_1_2"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:2_2_8_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>
<a NAME="E7:2_2_8_1_1_2"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">EMPTY</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E8:2_2_8_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E9:2_2_8_1_1_2"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:2_2_8_1_1_2"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">sumx</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E5:2_2_8_1_1"/><i><font color="Green">E86</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/><i><font color="Green">E87</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>
;<br/><a NAME="E7:2_2_8_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> implies <font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  );<br/><a NAME="E8:2_2_8_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i/>;<br/><a NAME="E9:2_2_8_1_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<b>assume </b><a NAME="E2:2_2_8_1_1_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:2_2_8_1_1_4"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:2_2_8_1_1_4"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:2_2_8_1_1_4"/><i><font color="Green">E95</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:2_2_8_1_1_4"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:2_2_8_1_1_4"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:2_2_8_1_1_4"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:2_2_8_1_1_4"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:2_2_8_1_1_4"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:2_2_8_1_1_4"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E12:2_2_8_1_1_4"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then </b><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/>
<b>hence </b><a NAME="E14:2_2_8_1_1_4"/>
<font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E10:2_2_8_1_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i><a NAME="E11:2_2_8_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E100</font></i>:  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> = 
<font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, </i>
<br/>.= 
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, </i>
<br/>.= 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><a NAME="E13:2_2_8_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:2_2_8_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:2_2_8_1_1"/><i><font color="Green">E102</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_2_8_1_1_6"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_6_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_1_6_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:2_2_8_1_1_6_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:2_2_8_1_1_6_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:2_2_8_1_1_6_1"/><b>then </b><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_6_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_6_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_1_6_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_6_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_6_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_1_6_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:2_2_8_1_1_6_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:2_2_8_1_1_6_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:2_2_8_1_1_6_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E7:2_2_8_1_1_6_1_1_2"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E8:2_2_8_1_1_6_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E9:2_2_8_1_1_6_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:2_2_8_1_1_6"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_6_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_6_2"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_6_2_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_6_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_1_6_2_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:2_2_8_1_1_6_2_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:2_2_8_1_1_6_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E7:2_2_8_1_1_6_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:2_2_8_1_1_6_2_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E9:2_2_8_1_1_6_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E10:2_2_8_1_1_6_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_6_2_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_6_2_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_6_2_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_1_6_2_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1"/>
( <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K1">0.</a>  or <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1"><b>case </b></a><a NAME="E1:2_2_8_1_1_7_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/><a NAME="E2:2_2_8_1_1_7_1_1"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc2.html#T34">MESFUNC2:34</a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_1_1_1_1"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:2_2_8_1_1_7_1_1_1_1_2"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E4:2_2_8_1_1_7_1_1_1_1_2"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">G</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:2_2_8_1_1_7_1_1"/><i><font color="Green">E119</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><a NAME="E5:2_2_8_1_1_7_1_1"/><i><font color="Green">E120</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/><a NAME="E6:2_2_8_1_1_7_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/><a NAME="E7:2_2_8_1_1_7_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E8:2_2_8_1_1_7_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:2_2_8_1_1_7_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E10:2_2_8_1_1_7_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:2_2_8_1_1_7_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/><a NAME="E12:2_2_8_1_1_7_1_1"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_1_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_2_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> =  <a href="supinf_2.html#K1">0.</a> ;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="supinf_2.html#K1">0.</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_1_2_1_1"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_2_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> = <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_1_2_1_2"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">b</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E14:2_2_8_1_1_7_1_1"/><i><font color="Green">E131</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>,<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>);<br/></i><a NAME="E15:2_2_8_1_1_7_1_1"/><i><font color="Green">E132</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E16:2_2_8_1_1_7_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E17:2_2_8_1_1_7_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E18:2_2_8_1_1_7_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E19:2_2_8_1_1_7_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E21:2_2_8_1_1_7_1_1"/><i><font color="Green">E136</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">m</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E22:2_2_8_1_1_7_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E23:2_2_8_1_1_7_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E24:2_2_8_1_1_7_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/><a NAME="E25:2_2_8_1_1_7_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:2_2_8_1_1_7_1_1_3"/><i><font color="Green">E139</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E140</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 2 = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E3:2_2_8_1_1_7_1_1_3"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:2_2_8_1_1_7_1_1_3"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">sumy</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E27:2_2_8_1_1_7_1_1"/><i><font color="Green">E142</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/><i><font color="Green">E143</font></i>: <font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><i><font color="Green">E144</font></i>:  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font> = 
<font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><a NAME="E30:2_2_8_1_1_7_1_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_1_6"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_1_6_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> ) or  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_6_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_1_6_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
( ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 ) )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_1_6_1_1_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><b>assume </b><a NAME="E2:2_2_8_1_1_7_1_1_6_1_1_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:2_2_8_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_2_8_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">k'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:2_2_8_1_1_7_1_1_6_1_1_1_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E10:2_2_8_1_1_7_1_1_6_1_1_1_1"/>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E11:2_2_8_1_1_7_1_1_6_1_1_1_1"/><b>then </b>
( <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:2_2_8_1_1_7_1_1_6_1_1_1_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:2_2_8_1_1_7_1_1_6_1_1_1_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:2_2_8_1_1_7_1_1_6_1_1_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E15:2_2_8_1_1_7_1_1_6_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E16:2_2_8_1_1_7_1_1_6_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_6_1_1_1_2"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>assume </b><a NAME="E2:2_2_8_1_1_7_1_1_6_1_1_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:2_2_8_1_1_7_1_1_6_1_1_1_2"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_2_8_1_1_7_1_1_6_1_1_1_2"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">k'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:2_2_8_1_1_7_1_1_6_1_1_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E10:2_2_8_1_1_7_1_1_6_1_1_1_2"/>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E11:2_2_8_1_1_7_1_1_6_1_1_1_2"/><b>then </b>
( <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:2_2_8_1_1_7_1_1_6_1_1_1_2"/><b>then </b>
<font color="Maroon">k'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:2_2_8_1_1_7_1_1_6_1_1_1_2"/><b>then </b>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E14:2_2_8_1_1_7_1_1_6_1_1_1_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E15:2_2_8_1_1_7_1_1_6_1_1_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E16:2_2_8_1_1_7_1_1_6_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_6_1_2"/>
(  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_1_6_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_1_7_1_1_6_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><i><font color="Green">E160</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, </i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>
;<br/><i><font color="Green">E161</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> = 
 <a href="finseq_1.html#K2">Seg</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E34:2_2_8_1_1_7_1_1"/>
( <font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9"><b>proof </b></a><div class="add">

<a NAME="E1:2_2_8_1_1_7_1_1_9"/><i><font color="Green">E162</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/>for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_1_9_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:2_2_8_1_1_7_1_1_9_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_1_9_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_9_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_1_7_1_1_9_1_1_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_1_9_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_1_7_1_1_9_1_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, , , <a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_2_8_1_1_7_1_1_9"/>
<font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>

<b>thus </b><a NAME="E3:2_2_8_1_1_7_1_1_9"/>
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:2_2_8_1_1_7_1_1_9"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_1_9_2"/><i><font color="Green">E165</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:2_2_8_1_1_7_1_1_9_2"/><b>then </b><i><font color="Green">E166</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:2_2_8_1_1_7_1_1_9_2"/><i><font color="Green">E167</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:2_2_8_1_1_7_1_1_9_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_2_8_1_1_7_1_1_9_2"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:2_2_8_1_1_7_1_1_9_2"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:2_2_8_1_1_7_1_1_9_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, , <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E9:2_2_8_1_1_7_1_1_9_2"/><b>then </b>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 
<b>by </b><i>, , , <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:2_2_8_1_1_7_1_1_9_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E11:2_2_8_1_1_7_1_1_9_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E13:2_2_8_1_1_7_1_1_9_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, , <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:2_2_8_1_1_7_1_1_9_2"/>
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> = 
 <a href="finseq_1.html#K2">Seg</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>

<b>thus </b><a NAME="E6:2_2_8_1_1_7_1_1_9"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_1_9_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_1_9_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:2_2_8_1_1_7_1_1_9_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_1_7_1_1_9_4"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E35:2_2_8_1_1_7_1_1"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D2">MESFUNC3:def 2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2"><b>case </b></a><a NAME="E1:2_2_8_1_1_7_1_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/><a NAME="E3:2_2_8_1_1_7_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E172</font></i>: <font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) );<br/><a NAME="E5:2_2_8_1_1_7_1_2"/><i><font color="Green">E173</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_2_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_2_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_2_1_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">v</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc2.html#T34">MESFUNC2:34</a></i>;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_2_2_1_1"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_2_1_2"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">v</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_2_2_1_2"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">G</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E7:2_2_8_1_1_7_1_2"/><i><font color="Green">E177</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><a NAME="E8:2_2_8_1_1_7_1_2"/><i><font color="Green">E178</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:2_2_8_1_1_7_1_2"/><b>then </b><i><font color="Green">E179</font></i>: 
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 2 )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon">EMPTY</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 2 as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/><a NAME="E11:2_2_8_1_1_7_1_2"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a>  ) );<br/><a NAME="E12:2_2_8_1_1_7_1_2"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>,<font color="Olive">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_2_3"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_2_3_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_3_1_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">v</font> =  <a href="supinf_2.html#K1">0.</a> ;<br/><b>reconsider </b><font color="Maroon">v</font> =  <a href="supinf_2.html#K1">0.</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_2_3_1_1"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a>  ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_3_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>set </b><font color="Maroon">v</font> =  <a href="mesfunc1.html#K10">1.</a> ;<br/><b>reconsider </b><font color="Maroon">v</font> =  <a href="mesfunc1.html#K10">1.</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_1_7_1_2_3_1_2"/>
( ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a>  ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">b</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E14:2_2_8_1_1_7_1_2"/><i><font color="Green">E185</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>,<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>);<br/></i><a NAME="E15:2_2_8_1_1_7_1_2"/><i><font color="Green">E186</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E16:2_2_8_1_1_7_1_2"/><b>then </b><i><font color="Green">E187</font></i>: 
( <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a>  )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E18:2_2_8_1_1_7_1_2"/><i><font color="Green">E188</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E19:2_2_8_1_1_7_1_2"/><i><font color="Green">E189</font></i>: 
( <font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:2_2_8_1_1_7_1_2_4"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E2:2_2_8_1_1_7_1_2_4"/><b>then </b>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_4"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/>

<a NAME="E4:2_2_8_1_1_7_1_2_4"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:2_2_8_1_1_7_1_2_4"/><b>then </b>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 2</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:2_2_8_1_1_7_1_2_4"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">sumy</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E21:2_2_8_1_1_7_1_2"/><i><font color="Green">E190</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/><i><font color="Green">E191</font></i>: <font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 0</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><i><font color="Green">E192</font></i>:  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">sumy</font> <a href="mesfunc1.html#K4">.</a> 1</span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><a NAME="E24:2_2_8_1_1_7_1_2"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_2_7"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_2_7_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> ) or  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_7_1_1"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_2_7_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
( ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 ) )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_2_7_1_1_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_7_1_1_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_2_7_1_1_1_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_7_1_1_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_7_1_1_1_2"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_2_7_1_1_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_7_1_1_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_7_1_2"/>
(  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_2_7_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_7_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E26:2_2_8_1_1_7_1_2"/>
( <font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8"><b>proof </b></a><div class="add">

<a NAME="E1:2_2_8_1_1_7_1_2_8"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E2:2_2_8_1_1_7_1_2_8"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E3:2_2_8_1_1_7_1_2_8"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E4:2_2_8_1_1_7_1_2_8"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<b>then </b><i><font color="Green">E196</font></i>:  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/>
<a NAME="E6:2_2_8_1_1_7_1_2_8"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/>for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_2_8_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:2_2_8_1_1_7_1_2_8_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_1_7_1_2_8_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_8_2_1_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_1_7_1_2_8_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, , , <a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a>, , <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_8_2_1_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_1_7_1_2_8_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_1_7_1_2_8_2_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:2_2_8_1_1_7_1_2_8"/>
<font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>

<b>thus </b><a NAME="E8:2_2_8_1_1_7_1_2_8"/>
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:2_2_8_1_1_7_1_2_8"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_2_8_3"/>
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:2_2_8_1_1_7_1_2_8_3"/><b>then </b>
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_8_3"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D8">MESFUNC1:def 8</a>, <a class="ref" href="extreal2.html#T20">EXTREAL2:20</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:2_2_8_1_1_7_1_2_8"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:2_2_8_1_1_7_1_2_8"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_1_7_1_2_8_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_1_7_1_2_8_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:2_2_8_1_1_7_1_2_8_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_1_7_1_2_8_4"/>
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>hence </b><a NAME="E27:2_2_8_1_1_7_1_2"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="mesfunc3.html#D2">MESFUNC3:def 2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:2_2_8_1_1"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2"/><i><font color="Green">E200</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2"/><i><font color="Green">E201</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> holds <br/><font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font>
 ;<br/>

<a NAME="E2:2_2_8_1_2_1"/><i><font color="Green">E203</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a1</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E3:2_2_8_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E204</font></i>: <font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc4.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc4.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="mesfunc4.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E6:2_2_8_1_2_1"/>
<font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:2_2_8_1_2"/><i><font color="Green">E205</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/><a NAME="E4:2_2_8_1_2"/><b>then </b><i><font color="Green">E206</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E5:2_2_8_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:2_2_8_1_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E7:2_2_8_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E8:2_2_8_1_2"/><b>then </b><i><font color="Green">E208</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">y</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E10:2_2_8_1_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E11:2_2_8_1_2"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>and </b><br/><a NAME="E12:2_2_8_1_2"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>and </b><br/><a NAME="E13:2_2_8_1_2"/><i><font color="Green">E212</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E14:2_2_8_1_2"/><i><font color="Green">E213</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E15:2_2_8_1_2"/><i><font color="Green">E214</font></i>: 
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="mesfunc3.html#D2">MESFUNC3:def 2</a>, , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  or ( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1"><b>case </b></a><a NAME="E1:2_2_8_1_2_2_1_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; ( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E2:2_2_8_1_2_2_1_1"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:2_2_8_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:2_2_8_1_2_2_1_1_1"/><i><font color="Green">E219</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_1_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_1_1_1"/><i><font color="Green">E222</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E8:2_2_8_1_2_2_1_1_1_1_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_1_1_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E4:2_2_8_1_2_2_1_1_1_1_2"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">H</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:2_2_8_1_2_2_1_1"/><i><font color="Green">E224</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>);<br/></i><a NAME="E5:2_2_8_1_2_2_1_1"/><i><font color="Green">E225</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:2_2_8_1_2_2_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E10:2_2_8_1_2_2_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_1"/><b>then </b><i><font color="Green">E226</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E13:2_2_8_1_2_2_1_1"/><i><font color="Green">E228</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> holds <br/><font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E14:2_2_8_1_2_2_1_1"/><i><font color="Green">E229</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E15:2_2_8_1_2_2_1_1"/><i><font color="Green">E230</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> holds <br/><font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_1_3"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_3"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:2_2_8_1_2_2_1_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">v</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_4"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_4_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> ) or  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green">E236</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_4_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_4_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_4_1_1_1_1"/><b>then </b><i><font color="Green">E238</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_4_1_1_1_1"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_4_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><i><font color="Green">E242</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E243</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><i><font color="Green">E244</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E12:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>then </b><i><font color="Green">E245</font></i>: <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v1</font>
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/><a NAME="E14:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E15:2_2_8_1_2_2_1_1_4_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E9:2_2_8_1_2_2_1_1_4_1_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_4_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_4_1_1_1_2"/><b>then </b><i><font color="Green">E247</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_4_1_1_1_2"/><b>then </b><i><font color="Green">E248</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E250</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><b>assume </b><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E254</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u1</font> )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E12:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>then </b><i><font color="Green">E256</font></i>: <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">u1</font>
<b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/><a NAME="E14:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E15:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E5:2_2_8_1_2_2_1_1_4_1_1_1_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">v1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:2_2_8_1_2_2_1_1_4_1_1_1_2_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_4_1_2"/>
(  not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_4_1_2"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_4_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E18:2_2_8_1_2_2_1_1"/><i><font color="Green">E258</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_2_8_1_2_2_1_1_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_5_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_1_5_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:2_2_8_1_2_2_1_1_5_1"/><i><font color="Green">E259</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_2_8_1_2_2_1_1_5_1"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:2_2_8_1_2_2_1_1_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:2_2_8_1_2_2_1_1_5_1"/><b>then </b><i><font color="Green">E261</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_5_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_1_1_1"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_1_1_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:2_2_8_1_2_2_1_1_5_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_5_1_1_2"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_5_1_1_2"/><b>then </b><i><font color="Green">E263</font></i>: 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_5_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="mesfunc4.html#E1:2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_5_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_5_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:2_2_8_1_2_2_1_1_5_1_1_2"/><i><font color="Green">E265</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:2_2_8_1_2_2_1_1_5_1_1_2"/><i><font color="Green">E266</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E14:2_2_8_1_2_2_1_1_5_1_1_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_5_1_1_2_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_1_1_2_1_1"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_1_1_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_1_1_2_1_1"/><b>then </b><i><font color="Green">E269</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_5_1_1_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:2_2_8_1_2_2_1_1_5_1_1_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E7:2_2_8_1_2_2_1_1_5_1_1_2_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:2_2_8_1_2_2_1_1_5"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:2_2_8_1_2_2_1_1_5"/><i><font color="Green">E272</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_2_8_1_2_2_1_1_5"/><i><font color="Green">E273</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_5_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_2_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_2_1"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_5_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_2_1_1_1"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_2_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_5_2_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_5_2_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_5_2_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E7:2_2_8_1_2_2_1_1_5_2_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_2_1_1_2"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_5_2_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_5_2_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E6:2_2_8_1_2_2_1_1_5_2_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_5_2_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_5_2_2"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_5_2_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b><i><font color="Green">E278</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_5_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_1_5_2_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E11:2_2_8_1_2_2_1_1_5_2_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><i><font color="Green">E279</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/><a NAME="E20:2_2_8_1_2_2_1_1"/><i><font color="Green">E280</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_7"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:2_2_8_1_2_2_1_1_7"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_7_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_7_1_1_1_1"/><b>then </b><i><font color="Green">E285</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_7_1_1_1_1"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_7_1_1_1_1"/><b>then </b><i><font color="Green">E287</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_7_1_1_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_7_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_7_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E8:2_2_8_1_2_2_1_1_7_1_1_1_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E290</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E7:2_2_8_1_2_2_1_1_7_1_1_1_2_1_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_7_1_1_1_2_1_2"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_7_1_1_1_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_7_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_7_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, </i>;<br/><b>hence </b><a NAME="E5:2_2_8_1_2_2_1_1_7_1_1_1_2_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_7_1_2"/><i><font color="Green">E292</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_7_1_2"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E295</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_7_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>hence </b><a NAME="E11:2_2_8_1_2_2_1_1_7_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E21:2_2_8_1_2_2_1_1"/><i><font color="Green">E296</font></i>: 
<font color="Maroon">H</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E22:2_2_8_1_2_2_1_1"/><i><font color="Green">E297</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:2_2_8_1_2_2_1_1"/><i><font color="Green">E298</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> holds <br/><font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_8"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_1_8"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:2_2_8_1_2_2_1_1_8"/><b>then </b><i><font color="Green">E300</font></i>: 
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E4:2_2_8_1_2_2_1_1_8"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_8_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_8_1_1"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_8_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_8_1_1"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_8_1_2"/><i><font color="Green">E303</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_8_1_2"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_1_8_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_8_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_8_1_2"/><b>then </b><i><font color="Green">E305</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E7:2_2_8_1_2_2_1_1_8_1_2"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, , , <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:2_2_8_1_2_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E25:2_2_8_1_2_2_1_1"/><b>then </b><i><font color="Green">E306</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E26:2_2_8_1_2_2_1_1"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_9"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_9_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_9_1_1"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_9_1_1"/><i><font color="Green">E309</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_9_1_1"/><i><font color="Green">E310</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_9_1_1"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:2_2_8_1_2_2_1_1_9_1_1"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_9_1_2"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_1_9_1_2"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_9_1_2"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E27:2_2_8_1_2_2_1_1"/><i><font color="Green">E312</font></i>: 
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x1</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_10"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_10_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x1</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_2_2_1_1_10_1"/><i><font color="Green">E314</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_10_1"/><i><font color="Green">E315</font></i>: 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_10_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_10_1"/><b>then </b><i><font color="Green">E316</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_10_1"/><b>then </b><i><font color="Green">E317</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_1_10_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_10_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_2_2_1_1_10_1_1_1"/><i><font color="Green">E318</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_1_10_1_1_1"/><i><font color="Green">E319</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_1_10_1_1_1"/><i><font color="Green">E320</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b><i><font color="Green">E321</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:2_2_8_1_2_2_1_1_10_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fl</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="measure1.html#T21">MEASURE1:21</a></i>;<br/><a NAME="E12:2_2_8_1_2_2_1_1_10_1_1_1"/>
<font color="Maroon">EMPTY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E13:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fl</font>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E14:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fl</font>
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E15:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E16:2_2_8_1_2_2_1_1_10_1_1_1"/><b>then </b>
<a href="supinf_2.html#K1">0.</a>  <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T42">EXTREAL1:42</a></i>;<br/><b>hence </b><a NAME="E17:2_2_8_1_2_2_1_1_10_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_1_10_1_1_2"/><i><font color="Green">E323</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">l</font> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>
;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_10_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_2_8_1_2_2_1_1_10"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_1_10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_8_1_2_2_1_1_10_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:2_2_8_1_2_2_1_1_10_2"/><b>then </b>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_1_10_2"/>
contradiction
 <b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2_2_8_1_2_2_1_1_10"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E28:2_2_8_1_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, </i>;<br/><b>then </b><font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>
;<br/><b>then </b><i><font color="Green">E324</font></i>:  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="convfun1.html#T7">CONVFUN1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="convfun1.html#T5">CONVFUN1:5</a></i>
;<br/> <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> = 
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">z</font>
<b>by </b><i>, , , <a class="ref" href="mesfunc3.html#D2">MESFUNC3:def 2</a>, , , , , <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a>, <a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x1</font>
<b>by </b><i>, , , , <a class="ref" href="mesfunc4.html#T1" target="_self">Th1</a>, , , </i>
<br/>.= 
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E32:2_2_8_1_2_2_1_1"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2"><b>case </b></a><a NAME="E1:2_2_8_1_2_2_1_2"/><i><font color="Green">E325</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R1">is_finite</a> 
 <b>by </b><i>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E327</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:2_2_8_1_2_2_1_2"/><i><font color="Green">E328</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E329</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E11:2_2_8_1_2_2_1_2"/><i><font color="Green">E330</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E12:2_2_8_1_2_2_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E13:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E331</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="mesfunc4.html#E3:2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E14:2_2_8_1_2_2_1_2"/><i><font color="Green">E332</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E4:2_2"><i><font color="Green">E38</font></i></a>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E15:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E333</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E5:2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/><a NAME="E16:2_2_8_1_2_2_1_2"/><i><font color="Green">E334</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_1"/><i><font color="Green">E335</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_1_1_1"/><i><font color="Green">E336</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green">E337</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E9:2_2_8_1_2_2_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_1_1_2"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_1_1_2"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E4:2_2_8_1_2_2_1_2_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">H</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E18:2_2_8_1_2_2_1_2"/><i><font color="Green">E339</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="mesfunc4.html#E8:2_2"><i><font color="Green">E42</font></i></a>);<br/></i><a NAME="E19:2_2_8_1_2_2_1_2"/><i><font color="Green">E340</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E20:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E341</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/><a NAME="E21:2_2_8_1_2_2_1_2"/><i><font color="Green">E343</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Olive">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_2_1_1"/><i><font color="Green">E344</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_2_2_1_2_2_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green">E345</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E4:2_2_8_1_2_2_1_2_2_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">c</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E23:2_2_8_1_2_2_1_2"/><i><font color="Green">E347</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>,<font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><a NAME="E24:2_2_8_1_2_2_1_2"/><i><font color="Green">E348</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E25:2_2_8_1_2_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E26:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E349</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E27:2_2_8_1_2_2_1_2"/><i><font color="Green">E350</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_3"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then </b> <a href="xboole_0.html#K1">{}</a>  = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E3:2_2_8_1_2_2_1_2_3"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E28:2_2_8_1_2_2_1_2"/><i><font color="Green">E351</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E29:2_2_8_1_2_2_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E30:2_2_8_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E31:2_2_8_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E32:2_2_8_1_2_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E33:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E352</font></i>: 
<font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E16:2_2"><i><font color="Green">E49</font></i></a>, </i>;<br/><i><font color="Green">E353</font></i>:  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E17:2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E35:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E354</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/><a NAME="E36:2_2_8_1_2_2_1_2"/><i><font color="Green">E355</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E37:2_2_8_1_2_2_1_2"/><i><font color="Green">E356</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> &amp; <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_5"/><i><font color="Green">E357</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_2_5"/><b>then </b><i><font color="Green">E358</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_5_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_5_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_5_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="mesfunc4.html#E7:2_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_5_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_5_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E15:2_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_5_1_2"/><b>then </b><i><font color="Green">E359</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_5_1_2"/><b>then </b><i><font color="Green">E360</font></i>: 
<font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_5_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_2_5_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E7:2_2_8_1_2_2_1_2_5_1_2"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) );<br/><a NAME="E38:2_2_8_1_2_2_1_2"/><i><font color="Green">E362</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>,<font color="Olive">v</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_6"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_6_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_6_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_6_1_1"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E3:2_2_8_1_2_2_1_2_6_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_6_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_6_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_6_1_2"/><b>then </b><i><font color="Green">E364</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><b>take </b>
<font color="Maroon">v</font>
;<br/><b>thus </b><a NAME="E4:2_2_8_1_2_2_1_2_6_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">z</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E40:2_2_8_1_2_2_1_2"/><i><font color="Green">E365</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>,<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i><a NAME="E41:2_2_8_1_2_2_1_2"/><i><font color="Green">E366</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E42:2_2_8_1_2_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E43:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E367</font></i>: 
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, </i>;<br/><a NAME="E44:2_2_8_1_2_2_1_2"/><i><font color="Green">E368</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E45:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E369</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E46:2_2_8_1_2_2_1_2"/><i><font color="Green">E370</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E47:2_2_8_1_2_2_1_2"/><i><font color="Green">E371</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E48:2_2_8_1_2_2_1_2"/><i><font color="Green">E372</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><i><font color="Green">E373</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E22:2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc4.html#E21:2_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E50:2_2_8_1_2_2_1_2"/><i><font color="Green">E374</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> holds <br/><font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_8"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_2_8"/><i><font color="Green">E376</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>]
 
<b>by </b><i>, , , , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_8_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_8_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_8_1_1"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_8_1_2"/><i><font color="Green">E377</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_8_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_8_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_8_1_2"/><b>then </b><i><font color="Green">E378</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_8_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E6:2_2_8_1_2_2_1_2_8_1_2"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E51:2_2_8_1_2_2_1_2"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_9"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_9_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> ) or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1"/><i><font color="Green">E381</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font>, <font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:2_2_8_1_2_2_1_2_9_1_1"/><i><font color="Green">E382</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_9_1_1"/><i><font color="Green">E383</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>] )
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_9_1_1"/><i><font color="Green">E384</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_1"/><i><font color="Green">E385</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E3:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b><i><font color="Green">E386</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E5:2_2_8_1_2_2_1_2_9_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E8:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E9:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E10:2_2_8_1_2_2_1_2_9_1_1_1"/><b>then </b><i><font color="Green">E387</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/>
<a NAME="E11:2_2_8_1_2_2_1_2_9_1_1_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_1_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_2_2_1_2_9_1_1_1_1"/><i><font color="Green">E388</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:2_2_8_1_2_2_1_2_9_1_1_1_1"/><i><font color="Green">E389</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:2_2_8_1_2_2_1_2_9_1_1_1_1"/><i><font color="Green">E390</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E9:2_2_8_1_2_2_1_2_9_1_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:2_2_8_1_2_2_1_2_9_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:2_2_8_1_2_2_1_2_9_1_1_1_1"/><b>then </b><i><font color="Green">E391</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:2_2_8_1_2_2_1_2_9_1_1_1_1"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E13:2_2_8_1_2_2_1_2_9_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/>
<a NAME="E14:2_2_8_1_2_2_1_2_9_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E15:2_2_8_1_2_2_1_2_9_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:2_2_8_1_2_2_1_2_9_1_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_2_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_2_1"/><i><font color="Green">E392</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_9_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_9_1_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:2_2_8_1_2_2_1_2_9_1_1_2_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_2_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_9_1_1_2_2"/><b>then </b><i><font color="Green">E393</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_9_1_1_2_2"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_2_2_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_2_2_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_9_1_1_2_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_1_2_2_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_9_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_2_9_1_1_2_2_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_2"/><i><font color="Green">E395</font></i>: 
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_9_1_2_1"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> or  not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_2_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_2_1_1"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_9_1_2_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_2_9_1_2_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_9_1_2_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_9_1_2_1_2"/>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_9_1_2_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_2_9_1_2_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/><a NAME="E53:2_2_8_1_2_2_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E54:2_2_8_1_2_2_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>;<br/><b>then </b><font color="Maroon">F1</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> = 
<font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
;<br/><b>then </b><i><font color="Green">E396</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/><i><font color="Green">E397</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E58:2_2_8_1_2_2_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_13"/><i><font color="Green">E398</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_2_13"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:2_2_8_1_2_2_1_2_13"/><b>then </b><i><font color="Green">E399</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>]
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_13_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_13_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_13_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_13_1_1"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_13_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_13_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_13_1_2"/><b>then </b><i><font color="Green">E400</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_13_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_13_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E5:2_2_8_1_2_2_1_2_13_1_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E59:2_2_8_1_2_2_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><b>then </b><i><font color="Green">E401</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/><i><font color="Green">E402</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a>, <a class="ref" href="mesfunc4.html#T2" target="_self">Th2</a></i>
;<br/><a NAME="E62:2_2_8_1_2_2_1_2"/><i><font color="Green">E403</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> holds <br/>for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_16"/><i><font color="Green">E404</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:2_2_8_1_2_2_1_2_16"/><b>then </b><i><font color="Green">E405</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>] )
 
<b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , </i>;<br/>
<a NAME="E3:2_2_8_1_2_2_1_2_16"/><i><font color="Green">E406</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:2_2_8_1_2_2_1_2_16"/><i><font color="Green">E407</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_16_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_16_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_16_1_1"/><i><font color="Green">E408</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_16_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_2_16_1_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_16_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_16_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b><i><font color="Green">E409</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b><i><font color="Green">E410</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_16_1_2"/><i><font color="Green">E411</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_16_1_2"/><i><font color="Green">E412</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, </i>;<br/><a NAME="E6:2_2_8_1_2_2_1_2_16_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b><i><font color="Green">E413</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E12:2_2_8_1_2_2_1_2_16_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:2_2_8_1_2_2_1_2_16_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><b>hence </b><a NAME="E15:2_2_8_1_2_2_1_2_16_1_2"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E63:2_2_8_1_2_2_1_2"/><i><font color="Green">E414</font></i>: 
<font color="Maroon">H</font>,<font color="Maroon">c</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E64:2_2_8_1_2_2_1_2"/><i><font color="Green">E415</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E65:2_2_8_1_2_2_1_2"/><i><font color="Green">E416</font></i>: 
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_17_1"/><i><font color="Green">E417</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> or  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_17_1_1"/>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> or  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_17_1_1_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_2_2_1_2_17_1_1_1"/><i><font color="Green">E418</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_17_1_1_1"/><i><font color="Green">E419</font></i>: 
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:2_2_8_1_2_2_1_2_17_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_2_17_1_1_1"/><b>then </b><i><font color="Green">E420</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_17_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_17_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_17_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/><b>then </b>
( 1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/><b>then </b><i><font color="Green">E421</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i/>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">Gk</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/>
<font color="Maroon">EMPTY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Gk</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">EMPTY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Gk</font>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/><b>then </b><i><font color="Green">E423</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Gk</font>
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E12:2_2_8_1_2_2_1_2_17_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="extreal2.html#T44">EXTREAL2:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_17_1_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_17_1_1_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_17_1_1_2"/><b>then </b>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E424</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_17_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E425</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_2_17_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fn1</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_2_17_1_1_2"/>
<font color="Maroon">EMPTY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Fn1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_2_17_1_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">EMPTY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fn1</font>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E427</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fn1</font>
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E12:2_2_8_1_2_2_1_2_17_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fn1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E13:2_2_8_1_2_2_1_2_17_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="extreal2.html#T44">EXTREAL2:44</a>, <a class="txt" href="mesfunc4.html#E6:2_2"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_17"/>
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/>


</div><b>end;</b></div><i><font color="Green">E428</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E67:2_2_8_1_2_2_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> holds <br/><font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_19"/><i><font color="Green">E429</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_19_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_19_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_19_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_2_8_1_2_2_1_2_19_1_1"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_19_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_19_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:2_2_8_1_2_2_1_2_19_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:2_2_8_1_2_2_1_2_19_1_2"/><b>then </b><i><font color="Green">E430</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_19_1_2"/><b>then </b><i><font color="Green">E431</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E432</font></i>: <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc4.html#E2:2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E6:2_2_8_1_2_2_1_2_19_1_2"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_2_19_1_2"/><b>then </b><i><font color="Green">E434</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, , , <a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E8:2_2_8_1_2_2_1_2_19_1_2"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, , , <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E9:2_2_8_1_2_2_1_2_19_1_2"/>
<font color="Maroon">z</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="mesfunc4.html#E19:2_2"><i><font color="Green">E59</font></i></a>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E68:2_2_8_1_2_2_1_2"/><b>then </b><i><font color="Green">E435</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><a NAME="E69:2_2_8_1_2_2_1_2"/><i><font color="Green">E436</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_20"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_2_8_1_2_2_1_2_20"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_2_2_1_2_20"/><i><font color="Green">E437</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x1</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:2_2_8_1_2_2_1_2_20"/><i><font color="Green">E438</font></i>: 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:2_2_8_1_2_2_1_2_20"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:2_2_8_1_2_2_1_2_20"/><b>then </b><i><font color="Green">E439</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:2_2_8_1_2_2_1_2_20"/><b>then </b><i><font color="Green">E440</font></i>: 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>
<a NAME="E10:2_2_8_1_2_2_1_2_20"/><i><font color="Green">E441</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E11:2_2_8_1_2_2_1_2_20"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Fk</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_2_8_1_2_2_1_2_20_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_20_1_1"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_20_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2_8_1_2_2_1_2_20_1_1"/><i><font color="Green">E443</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:2_2_8_1_2_2_1_2_20_1_1"/><i><font color="Green">E444</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E5:2_2_8_1_2_2_1_2_20_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:2_2_8_1_2_2_1_2_20_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E7:2_2_8_1_2_2_1_2_20_1_1"/><b>then </b><i><font color="Green">E445</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:2_2_8_1_2_2_1_2_20_1_1"/>
<font color="Maroon">EMPTY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Fk</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E9:2_2_8_1_2_2_1_2_20_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">EMPTY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fk</font>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E10:2_2_8_1_2_2_1_2_20_1_1"/><b>then </b><i><font color="Green">E446</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fk</font>
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E11:2_2_8_1_2_2_1_2_20_1_1"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E12:2_2_8_1_2_2_1_2_20_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_2_8_1_2_2_1_2_20_1_2"><b>suppose </b></a><a NAME="E1:2_2_8_1_2_2_1_2_20_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then </b> <a href="supinf_2.html#K1">0.</a>  = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E3:2_2_8_1_2_2_1_2_20_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E70:2_2_8_1_2_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, </i>;<br/><b>then </b><i><font color="Green">E447</font></i>: <font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>
<br/>.= 
<font color="Maroon">x1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , </i>
;<br/> <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> = 
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">z</font>
<b>by </b><i>, , , <a class="ref" href="mesfunc3.html#D2">MESFUNC3:def 2</a>, , , <a class="txt" href="mesfunc4.html#E18:2_2"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="mesfunc4.html#E13:2_2"><i><font color="Green">E43</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="convfun1.html#T7">CONVFUN1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p5">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">H</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="convfun1.html#T5">CONVFUN1:5</a>, <a class="txt" href="mesfunc4.html#E1:2_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="mesfunc4.html#T1" target="_self">Th1</a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">x1</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="convfun1.html#T5">CONVFUN1:5</a></i>
<br/>.= 
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="ref" href="convfun1.html#T7">CONVFUN1:7</a></i>
;<br/><b>hence </b><a NAME="E73:2_2_8_1_2_2_1_2"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:2_2_8_1_2"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E24:2_2"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">f</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">F</font> being  <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font><br/> for <font color="Olive">a</font>, <font color="Olive">x</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>   st <font color="Olive">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) &amp; <font color="Olive">F</font>,<font color="Olive">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Olive">f</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">F</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">x</font> holds <br/><font color="Olive">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Olive">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> ) &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> holds <br/> <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Olive">x</font>
 ;<br/>


</div>
